Measuring a subset of an interval with a basic property (the couch potato tale) This is a basic illustration of a problem I need to then generalize. Below the problem and a little story that should clarify what I have in mind. I hope this is clear enough.
Consider the $[0,1]$ interval on the real line. Let $X$ be the set of all subsets of $[0,1]$ of length $1/4$ (length obtained using a uniform-like probability measure or Lebesgue measure if you prefer). What is the fraction of these subsets that has a non-empty intersection with the subset $[0,1/3]$?
Consider the following stylized problem. Each day a broadcasting station transmits a mixture of programming and advertising. On average, $2/3$ of the total airtime is actual content while the residual $1/3$ is advertising messages. There is an individual who watches TV from time to time at random times. That could happen in one unique session or in two sessions, three sessions and so on. Even infinitely many sessions. The only thing that we know about the individual is that he is in front of the TV a quarter of a day in total. Given this information what is the probability that this individual avoids all commercials? 
 A: As Brian M. Scott has explained you would need a probability measure on the set $X$ of all "admitted" subsets of $[0,1]$. When the admitted subsets are just intervals with randomly chosen endpoints it is not to difficult to establish such a measure and then to compute the probability that two randomly chosen intervals intersect.
On the other hand, when the collection of admitted sets is large then the probability you are envisaging will be tiny. Look at the following toy model:
A day of 24 hours is partitioned into 1440 intervals of one minute. The broadcaster selects each interval with probability ${1\over3}$ for advertising, and the viewer selects each interval with probability ${1\over4}$ for viewing. Assuming that all these selections are done independently the probability that a given interval is selected by both is ${1\over12}$, and the probability that no interval is selected by both then amounts to
$$\left({11\over12}\right)^{1440}\doteq 3.841\cdot 10^{-55}\ .$$
A: In my simple-minded way,
being ignorant of most of measure theory,
I would say that
the left endpoint of the interval
has to be at least
$0$ and at most $\frac13$.
Since the leftmost point
is at most $\frac34$,
the proportion is
$\frac{1/3}{3/4}
=\frac{4}{9}
$.
Now,
please explain to me
what I am not understanding.
